Average Error: 35.8 → 27.9
Time: 9.4s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.0997944018617001:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.0997944018617001:\\
\;\;\;\;\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double f(double x, double y) {
        double r691560 = x;
        double r691561 = y;
        double r691562 = 2.0;
        double r691563 = r691561 * r691562;
        double r691564 = r691560 / r691563;
        double r691565 = tan(r691564);
        double r691566 = sin(r691564);
        double r691567 = r691565 / r691566;
        return r691567;
}
↓
double f(double x, double y) {
        double r691568 = x;
        double r691569 = y;
        double r691570 = 2.0;
        double r691571 = r691569 * r691570;
        double r691572 = r691568 / r691571;
        double r691573 = tan(r691572);
        double r691574 = sin(r691572);
        double r691575 = r691573 / r691574;
        double r691576 = 1.0997944018617;
        bool r691577 = r691575 <= r691576;
        double r691578 = 1.0;
        double r691579 = cos(r691572);
        double r691580 = r691578 / r691579;
        double r691581 = 1.0;
        double r691582 = r691577 ? r691580 : r691581;
        return r691582;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.8
Target29.0
Herbie27.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.23036909113069936 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.1028524068119138 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 1.0997944018617

    1. Initial program 21.3

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
    2. Using strategy rm

    3. Applied tan-quot21.3

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity21.3

      \[\leadsto \frac{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{1 \cdot \cos \left(\frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

    6. Applied *-un-lft-identity21.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \sin \left(\frac{x}{y \cdot 2}\right)}}{1 \cdot \cos \left(\frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

    7. Applied times-frac21.3

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

    8. Applied associate-/l*21.3

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}}}}\]

    9. Simplified21.3

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]

    if 1.0997944018617 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

    1. Initial program 60.9

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

    2. Taylor expanded around 0 39.1

      \[\leadsto \color{blue}{1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification27.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.0997944018617001:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.2303690911306994e+114) 1 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1))

  (/ (tan (/ x (* y 2))) (sin (/ x (* y 2)))))